lemmathmplain If \(X=\mathcal {M}_O(n,a,b)\)  is feasible and \(\mathcal {R},\mathcal {B}\mbox{ and }\mathcal {G}\)  are as described above,
then the edge orbits of \(\mathrm {Aut}(X)\)  are one of the following:
(i) \(\mathcal {R},\mathcal {B}\mbox{ and }\mathcal {G}\) ;
(ii) \(\mathcal {R}\cup \mathcal {B}\mbox{ and }\mathcal {G}\) ;
(iii) \(\mathcal {R}\cup \mathcal {G}\mbox{ and }\mathcal {B}\) ; or
(iv) \(\mathcal {R}\cup \mathcal {B}\cup \mathcal {G}\)  in which case \(X\)  is arc-transitive.
Proof. Because the orbits of a permutation group are defined as the equivalence classes
of an equivalence relation and \(\mathcal {R},\mathcal {B}\mbox{ and }\mathcal {G}\)  are the
edge orbits of \(G\) , it follows that the orbits of \(\mathrm {Aut}(X)\)  are a disjoint collection
of unions of \(\mathcal {R},\mathcal {B}\mbox{ and }\mathcal {G}\) . It suffices to show that
\(\mathcal {B}\cup \mathcal {G}\)  and \(\mathcal {R}\)  may not occur.
If the edge orbits of \(\mathrm {Aut}(X)\)  are \(\mathcal {B}\cup \mathcal {G}\)  and \(\mathcal {R}\) ,
then there must be an automorphism mapping the edge \([u_0,u_1]\)  onto the edge \([u_{n-1},u_0]\) .
Both ways this may happen force \(b=n-a\)  which violates \(X\)  being feasible.
Finally, if condition (iv) happens, then Tutte [1]} proved that a trivalent graph that is both
vertex-transitive and edge-transitive also is arc-transitive. 
We break the considerations for this class of graphs into four cases. It is easy to see that
\(\mathcal {M}_O(n,a,b)\)  has girth 4 if and only if \(a=1\)  or \(b-a=n/2\) . We need not consider
the latter case because \(\mathrm {gcd}(n,n/2)=2\)  implies that \(n=4\)  forcing the graph to be
\(\mathrm {GP}(4,1)\) . Thus, we consider the case of \(a=1\) 
in this section. Moreover, the 4-cycles of \(\mathcal {M}_O(1,b)\)  form a 2-factor. The next
result describes some useful block systems for \(\mathrm {Aut}(\mathcal {M}_O(1,b))\) , but a
definition is required first. Note that the edge \([v_0,v_1]\)  is blue because \(a=1\) .
We introduce an auxiliary graph \(Y\)  to describe the first block system.
Let \(Y\)  have the same vertex set as \(X\) . The edges of \(Y\)  are the edges of \(\mathcal {G}\) 
together with the diameter edges of the 4-cycles, that is, given the 4-cycle \([u_i,u_{i+1},v_{i+1},v_i,u_i]\)  in \(X\) , \(i\)  even, then \([u_i,v_{i+1}]\)  and \([u_{i+1},v_i]\)  are edges of \(Y\) . The
graph \(Y\)  is regular of valency 2 so that it consists of vertex-disjoint cycles. Let A
denote the collection of vertex sets of the cycles comprising \(Y\) . The elements of A
are called anchor chains of \(X\) .
Let F denote the collection of vertex sets of the 4-cycles, that is, the vertex sets of the
subgraph composed of the edges of \(\mathcal {R}\cup \mathcal {B}\) . The edges of
\(\mathcal {G}\)  form a perfect matching in \(X\) . Let M be the collection of 2-subsets of
ends of the edges in this perfect matching.
